Draw a horizontal line on a plain sheet of paper. Call the line "x." Segment the line into 10 equally spaced sections with each section separated by small vertical hash marks. Label the hash marks 1 through 10.
Draw a vertical line starting at the same point you originated the horizontal line. Call this line y. Divide the line into 10 equally spaced horizontal marks and label them from 10 to 100, with each hash mark representing an increment of 10.
Calculate the values of y = x^2 + 3x + 1 and plot the values on a graph. Start with x = 1. At x = 1, y = 5. On the x-line, locate the hash mark labeled 1. While at 1 hash mark, go up vertically to the 5 hash mark on the y line and place a dot at that point. At x = 2, y = 11. Locate the 2 hash mark on the x-line and go up vertically to the approximate 11 mark on the y line and place a dot at that point. At x = 3, y = 19. Locate the 3 hash mark on the x-line and go up vertically to the approximate 19 mark on the y line and place a dot at that point. Repeat this process x = 4 through x = 10.
Connect all of the dots on the graph by drawing a line from on to the other. The picture you see is the graph of the polynomial y = x^2 + 3x + 1.